To set up the integral representing the surface area obtained by rotating the parametric curve given by $x = \sin^2(t)$, $y = \sin(2t)$ about the $x$-axis, we'll use the surface area formula for a parametric curve rotated about the $x$-axis:

$A = \int_{a}^{b} 2\pi y \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

Given:

• $x = \sin^2(t)$
• $y = \sin(2t)$
• The parameter range is $0 \leq t \leq \frac{\pi}{2}$

Step 1: Compute $\frac{dx}{dt}$ and $\frac{dy}{dt}$

• $\frac{dx}{dt} = \frac{d}{dt} \left( \sin^2(t) \right) = 2\sin(t)\cos(t) = \sin(2t)$
• $\frac{dy}{dt} = \frac{d}{dt} \left( \sin(2t) \right) = 2\cos(2t)$

Step 2: Set up the surface area integral

Substitute the values of $y$, $\frac{dx}{dt}$, and $\frac{dy}{dt}$ into the surface area formula:

$A = \int_{0}^{\frac{\pi}{2}} 2\pi \sin(2t) \sqrt{\left( \sin(2t) \right)^2 + \left( 2\cos(2t) \right)^2} \, dt$

Step 3: Simplify the square root expression

The term inside the square root simplifies as follows:

$\left( \sin(2t) \right)^2 + \left( 2\cos(2t) \right)^2 = \sin^2(2t) + 4\cos^2(2t)$

Thus, the integral becomes:

$A = \int_{0}^{\frac{\pi}{2}} 2\pi \sin(2t) \sqrt{\sin^2(2t) + 4\cos^2(2t)} \, dt$

This is the integral representing the surface area of the curve rotated about the $x$-axis.

Would you like me to proceed with further simplification or evaluation of this integral?

Related questions:

1. What is the physical significance of the surface area obtained by rotating a curve?
2. How does the parametric representation affect the surface area calculation for rotational surfaces?
3. Can surface area integrals be evaluated for curves rotated around the $y$-axis instead of the $x$-axis?
4. How would you handle the surface area if the limits of the parameter $t$ were different?
5. How does the shape of the curve affect the complexity of the surface area integral?

Tip:

When solving problems involving surface areas of rotation, always check if there's symmetry that can simplify the integral before proceeding with full evaluation.